Why do I "get embarrassed being out with people with public school nicknames."?

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Why do I "get embarrassed being out with people with public school nicknames."?

The quoted part is part of a line I found in a British TV series "Miranda".And the context may be seen as follow:
"I know. But as Tilly says,when you're dumped in a boarding school dorm aged nine,you'll bond for life,even if you hate each other.Or in my case, get embarrassed being out with people with public school nicknames.In my class, Milly, Tilly,Bella, Bunty, Hootie,Pussy, Puggle and Podge." Are these nicknames only used in public schools ?
Apart from the reason why the speaker "get embarrassed being out with people with public school nicknames.",I will be very very much obliged if you could also help me to understand the connotation of each nickname stated in the line.I'm anxious for your help please.